04/19/23 Steve King, Coseners House 1

Models of Neutrino Mass

•Introduction (neutrino masses,mixings)

•Theory of neutrino mass (Majorana,Dirac,See-saw,Majorana matrices)

•Natural hierarchy (sequential right-handed neutrino dominance)

•Relation between neutrino factory CP violation and leptogenesis?

•SUSY and lepton flavour violation: see-saw vs. SUGRA

•GUTs and Family Symmetry

04/19/23 Steve King, Coseners House 2

Possible three neutrino mass patterns with LMA and .

21m

22m

23m

23m

22m21m

232m

232m

221m

221m

“Normal” “Inverted”

232 0m 2

32 0m

2(0.00 )eV 2(0.10 )eV 2(0.20 )eV

2 232| | (0.05 )m eV 2 2

21 (0.008 )m eV

2(0.23 )eV 2(0.00 )eV

04/19/23 Steve King, Coseners House 3

126

4 23

2. 0 13

Large Mixing Angles

1

23 13 12 2

3

e

R R R

atmospheric

solar LMA MSW

CHOOZ

04/19/23 Steve King, Coseners House 4

The Neutrino Mixing Matrix U

†0 0

0 0

0 0

L R

eE EE

LR

m

V m V m

m

†L LEU V V

23 13 12 12U R U R P

23 23 23

23 23

1 0 0

0

0

R c s

s c

13 13

13

13 13

0

0 1 0

0

i

i

c s e

U

s e c

12 12

12 12 12

0

0

0 0 1

c s

R s c

Parametrising

Constructing

Majorana matrix

3

2

1

00

00

00

m

m

m

VmVT

LLLL

1

212

0 0

0 0

0 0 1

i

i

e

P e

04/19/23 Steve King, Coseners House 5

In the Standard Model neutrinos are massless, and a neutrino and anti-neutrino are distinguished by a (total)

conserved lepton number L.

cLLLLm

Origin of neutrino mass

Majorana mass

Dirac mass LR L Rm

Majorana or Dirac?

(conserves L)

CP conjugate of left-handed

neutrino

Right-handed neutrinos

cRR R RM

from Yukawa couplings

(violates L)

04/19/23 Steve King, Coseners House 6

See-saw mechanism

1 TLL LR RR LRm m M m

cLL L Lm

0 cLRc L

L R TLR RR R

m

m M

Diracmatrix

Heavy Majorana matrix

Light Majorana matrix

Diagonalise

04/19/23 Steve King, Coseners House 7

0 0

0 1 12

0 1 1

0HILL

mm

( )

1 1

1 0 02

1 0 0

0IH ALL

mm

Inverted hierarchy

Type A (zero in 11)

( )

0 0

0 1 2 1 2

0 1 2 1 2

1IH BLLm m

Degenerate Pseudo-Dirac

Type B (non-zero 11)

( 2)

0 0

0 0 1

0 1 0

1DEG BLLm m

( 1)

0 0

0 1 0

0 0 1

1DEG BLLm m

( )

1 1

2 21 1 1

2 221 1 1

2 22

0

DEG ALLm m

Large neutrinoless double beta decay (Petcov,…)

Hierarchy

Leading order consistent with LMA MSW Barbieri,Hall,Smith,Strumia,Weiner;Altarelli,Feruglio;…

Natural cases: SFK hep-ph/0204360

2 2 21 2 3m m m

2 2 21 2 3m m m

2 2 21 2 3m m m

LLm

04/19/23 Steve King, Coseners House 8

Natural Neutrino Hierarchy?

Natural eigenvalues are why ?

Technically need a small 23 sub-determinant:

222 23

23 33

detm m

mm m

2 3m m

But why should the sub-determinant be small ?

How can we get this from the see-saw mechanism?

0 0

0 1 12

0 1 1

0HILL

mm

2 22 3m m 2 2

2 3m m

04/19/23 Steve King, Coseners House 9

0 0

0 0

0 0RR

Y

X

M X

LR

a a d

m b b

c c

e

f

2 2 2

21

2 2

2

2

2

.

. .

TLL LR RR LR

a a d a b ab d a c ac d

X X X X X X

b b b c bcm m M m

X X X X

e f

Y Y Y

e ef

Y Y

c

X

fc

YX

If one right-handed neutrino of mass Y dominates then sub-determinant is naturally small: single right-handed

neutrino dominance (SFK PLB 98, NPB 99)

See-saw with diagonal heavy Majorana matrix

0 0

0 0

0 0RRM X

X

Y

'LR

d a a

m b b

c c

e

f

Columns can be re-ordered without loss of generality

2 2 | | | |,

, , , , ,

|

, ,

| ,| | ,| | xy x y

X Xx y a b c x

e f e

y cY

a

f

b

Sequential dominance (SFK NPB2000)

04/19/23 Steve King, Coseners House 10

| || | || e fd

Analytic estimates for neutrino masses and mixing angles (valid for sequential dominance)

SFK hep-ph/0204360

2 2 | | | |,

, , , , ,

|

, ,

| ,| | ,| | xy x y

X Xx y a b c x

e f e

y cY

a

f

b

23t|

an|

| |

e

f

1223 23

| |tan

| | cos( ) | | cos( )b c

a

c b s c

2

3

2| | | |m

e f

Y

1 2 3m m m

Sequential dominance

2

2 212

| |am

Xs

1 ( )x y

m OX

* *( )

13 3/ 22 2

| | ( )

| | | |a ei a e b f c Y

ee f X

23 23| | sin( ) | | sin( )b cc b s c

b b a c c a e f

* *arg( )e a e b f c

(set d=0 below)

Phases fixed by real 12 angle condition

and is fixed by the real 13 angle condition

where

Mass hierarchy and large

mixing angles naturally!

04/19/23 Steve King, Coseners House 11

0 0

0 0

0 0RR

Y

X

M X

0 0 0

0 0

0

1

10LR

a a d

Y b b

fc c

e

Light or Heavy Sequential Dominance?

0 0

0 0

0 0RRM X

X

Y

0 0 0

0 0 0

' 0 0 1LR

d a a

Y b b

c

e

f c

Light Sequential Dominance (LSD)

Heavy Sequential Dominance (HSD)

Y X X

X X Y

“Lop-sided” (large 23 element)

“Quark-like” (no large off-diagonal

entries)Note that X’ is irrelevent for neutrino masses, mixings (and leptogenesis-see later), and the model reduces to an effective 2 right-handed neutrino

model SFK 2000; Frampton,Glashow,Yanagida 2002; Raidal,Strumia 2002.

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•Right-handed neutrinos are produced in early universe and decay out of equilibrium giving net lepton number L

•CP violation from complex Yukawa couplings

•L is processed into B via B-L conserving sphalerons

Fukugita,Yanagida; Buchmuller,Plumacher; Di Bari, Davidson…

10 *110 /B

baryonsY d g

photons

Leptogenesis

† 21 12

1 †2 11

Im{[( ) ] }3

8 ( )LR LR

LR LR

M Y Y

M Y Y

1 1

11 1

( ) ( )

( ) ( )R j U R j U

R j U R j U

L H L H

L H L H

1R 1R†

jLjL

UHUH

UH

iL

2R

2R1R

Relation between CP violation at a neutrino factory and in leptogenesis? SFK

hep-ph/0211228

0 0

0 0

0 0RRM X

X

Y

0

1LR e

f

a a

Y b b

c

Heavy RH neutrino of mass X’ is irrelevant for both leptogenesis and neutrino masses, mixings

† 212Im{[( ) ] } sin 2B LR LR COSMOY Y Y * *arg( )COSMO e b f c

23 2 23 3

23 2 23 3

| | sin | | sintan( )

| | cos | | cosCOSMO

b s c c

b s c c

23 2 23 3

23 2 23 3

| | sin | | sintan( )

| | cos | | cosCOSMO

b c c s

b c c s

2 3,b e c f Neutrino factory phase and leptogenesis phase are linked in terms of two invariant see-saw

phasesRemarkably the leptogenesis phase is equal to

the phase which enters neutrinoless double beta decay !

0| | | |COSMO

Assume 11 texture zero (see

later)

12' 10X X GY eV LSD

Di Bari,Buchmuller,Plumacher

In HSD (Hirsch,SFK) and gravitino bound may be respected (but no leptogenesis-MNS link since X’ couplings are relevant for

leptogenesis but not neutrino masses, mixings)

9' 10X X GeY V

04/19/23 Steve King, Coseners House 14

SUSY and Lepton Flavour Violation

If SUSY is present then neutrino masses inevitably lead to lepton flavour violation due to radiatively generated off-diagonal slepton

masses

Borzumati, Masiero; Hisano, Moroi, Tobe, Yamaguchi; SFK, Oliveira; Casas, Ibarra; Lavignac, Masina, Savoy;Blazek,SFK

0 0 0

0

1

0 1

0 0

HSD/lopsided will lead to large due to

large 23 Yukawa.Blazek, SFK

32 ln ln lnGUT GUT GUTM M MC b c bc ef

X X Y

32

32 2 2

32 22( ) ( , , ) | | tan

LF

BR f M m mG

32

2 22( ) 0 0

322

(3 )

8LLA

L

m Am C

LR e

a a d

Y b b

c c f

0 0

0 0

0 0RR

Y

X

M X

( )BR

2 2 2 2†0 0

2

0

(3 )

16LR

L LLR LR

Y

dm dm m AY Y

dt dt

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R DHUH W W L

23

2m

2M2 ug v

2gh

04/19/23 Steve King, Coseners House 16

3 14

0 0 0

0 0 3.10

0 0 1

RRM GeV

212

222

232

1

1

0

LR t

a

Y a h

a

Lepton Flavour Violation in Constrained Minimal Supersymmetric Standard Model

6

3 14

0 0

0 0 3.10

0 0 1

RRM GeV

212

3

3

222

232

0

1

LR t

a

Y a h

a221232

0.15m

m

Blazek,SFK hep-ph/0211368

Large 23 element (lop-sided)

HSD parametrisation with dominant heaviest RH neutrino in

33 position

LSD parametrisation with dominant lightest RH neutrino in

11 position

04/19/23 Steve King, Coseners House 17

Blazek,SFK hep-ph/0211368

tan 50

13 angle depends on a ratio of

Yukawa couplings

32

22

a

a

HSD

04/19/23 Steve King, Coseners House 18

HSD predictions for

constrained MSSM

32

22

tan 50,

1.

a

ra

61.1 10 (90% . .)C L 111.2 10 (90% . .)C L

Blazek,SFK hep-ph/0211368

04/19/23 Steve King, Coseners House 19

LSD predictions for constrained

MSSM

32

22

tan 50,

1.

a

ra 61.1 10 (90% . .)C L 111.2 10 (90% . .)C L

Blazek,SFK hep-ph/0211368

04/19/23 Steve King, Coseners House 20

Supergravity LFV can mimic see-saw LFV

Iain Peddie, SFK (preliminary)

04/19/23 Steve King, Coseners House 21

t

b

cs

ud

e

The goal is to describe the quark and lepton mass hierarchies

Charge +2/3 quarks

Charge –1/3 quarks

Charged leptons

A neutrino hierarchy

GGUTGFamily

1

2

3

310

1

110210

110

110

2101210

310410

1210

1110

04/19/23 Steve King, Coseners House 22

GUTs

E 6

(5) (1)SU U (3) (3) (3)C L RSU SU SU

(4) (2) (2)PS L RSU SU SU

(3) (2) (2) (1)C L R B LSU SU SU U

(3) (2) (1)C L YSU SU U

(5)SU

(10)SO

04/19/23 Steve King, Coseners House 23

Family Symmetry

U(1)

SU(2)

SU(3)

SO(3)

S(3)Nothing

Democracy, Fritzsch,…Hierarchy

Froggatt,Nielsen,Ross…

1 1 1

1 1 1

1 1 1

demm

0 0 0

0 0 0

0 0 1

hierm

(3) (3)L RO O

(3) (3)L RS S

Harrison, Scott

04/19/23 Steve King, Coseners House 24

G.Ross, SFK

G.Ross, Velasco-Sevilla

3 3

3 2 22 23 3

3 223

0

1

uY

6

23

6

3 0 0

0 0

0 0 1RRM M

• Allows a universal form for mass matrices

• Example of Light Sequential Dominance (LSD)

• Texture zero in 11 position leptogenesis-neutrino factory link

Leads to approximately symmetric mass matrices

(3) (10)SU SO

3 3

3 2 2

3 2

0

1

dY

3 3

3 2 2

3 2

0

3 3

3 1

eY

3 3

3 2 2

3 2

0

1

Y

0.05, 0.15

Flavour Unification

04/19/23 Steve King, Coseners House 25

Conclusion• Small neutrino masses can be elegantly explained by see-saw

mechanism• Sequential dominance then provides a natural explanation of a

neutrino mass hierarchy and large mixing angles• The dominant right-handed neutrino may be either the heaviest (HSD)

or the lightest one (LSD)• LSD with 11 zero links leptogenesis phase and the CP phase which is

measurable at a neutrino factory, but is inconsistent with SUSY models• HSD has no such link, but may be consistent with SUSY models• SUSY leads to lepton flavour violating processes, which can

discriminate HSD from LSD• SUGRA LFV can mimic see-saw LFV• GUTs and family symmetry can solve flavour problem G.Ross talk